GFCP - Polygons Lesson

Polygons

What are polygons?  A polygon is a two dimensional, closed figure formed by three or more segments. Each side intersects exactly two sides, one at each endpoint.

The “corner” is called a vertex (vertices – plural). The segments are called sides.

Polygon Activity

 

Convex vs. Concave

You can determine if a polygon is convex or concave by extending all the sides. If the extended line lies inside the polygon, then it is concave. If the extended sides are outside the polygon, then it is convex. Let’s look at figure A, C and D from the previous example.

Polygon Properties

Before we can dive into proving geometric theorems, we need to be reminded of the properties of some common polygons. Please spend some time studying the table below. You can take a screenshot of this table and print it out or copy them into your notes to help you study.  

Properties of Polygons

Polygon

Image

Properties

Prove by...

Formulas used

Parallelogram

 

image of parallelogram depcting angles and side lengths
  1. opposite sides are parallel
  2. opposite sides are congruent
  3. opposite angles are congruent
  4. consecutive angles are supplementary (add to 180 degrees)
  5. diagonals bisect each other

Prove one of the following:

  1. both pairs of opposite sides are parallel
  2. both pairs of opposite sides are congruent
  3. one pair of opposite sides are parallel and congruent

4. diagonals bisect each other

1. slope formula

2. distance formula

3. slope & distance formula

4. midpoint formula

Rectangle

image of rectangle depicting angles and side lengths
  1. opposite sides are parallel
  2. opposite sides are congruent
  3. all angles are right angles (90 degrees)
  4. diagonals bisect each other
  5. diagonals are congruent

 

Find the slope of all 4 sides.

Show opposite sides are parallel (have same slope) and show consecutive sides are perpendicular (have opposite reciprocal slope).

Slope formula

Rhombus

image of rhombus depicting angles and side lengths
  1. opposite sides are parallel
  2. all sides are congruent
  3. opposite angles are congruent
  4. consecutive angles are supplementary
  5. diagonals are perpendicular and bisect each other
  6. diagonals bisect the angles

Show all sides are congruent

Distance formula

Square

image of square depicting angles and side lengths
  1. opposite sides are parallel
  2. all sides are congruent
  3. all angles are right angles (90 degrees)
  4. diagonals are perpendicular and bisect each other
  5. diagonals bisect the angles
  6. diagonals are congruent

Show all 4 sides are congruent and show consecutive sides are perpendicular (or diagonals are congruent)

Distance formula and slope formula

Trapezoid

image of trapezoid
  1. bases are parallel
  2. legs are not parallel
  3. each lower base angle is supplementary to the corresponding upper base angle

Show 1 pair of opposite sides are parallel (same slopes) and show other pair of opposite sides are NOT parallel (different slopes)

Slope formula

Right Triangle

image of right triangle depicting the 90° angle and labelled ABC
  1. one angle is 90 degrees (right angle)
  2. the two acute angles are complementary (add to 90 degrees)
  3. The Pythagorean theorem holds true (a2+b2=c2)

Prove one of the following:

1. Use distance formula to find the length of each side & show Pythagorean Theorem holds true

2. Use slope formula to find the slope of each side & show that there is one pair of sides that have opposite reciprocal slopes (forming a right angle)

1. distance formula & Pythagorean Theorem

2. slope formula

Scalene Triangle

image of scalene triangle depicting angles and side lengths

Triangle in which all 3 sides are different lengths

Use distance formula to find the length of all 3 sides. Show that no two sides are equal

Distance formula

Isosceles Triangle

 image of isosceles triangle depicting angles and side lengths

Triangle in which exactly 2 sides have same length

Use distance formula to find the length of all 3 sides. Show that two sides are equal

Distance formula

Equilateral Triangle

 image of equilateral triangle depicting angles and side lengths

Triangle in which all 3 sides are the same length

Use distance formula to find the length of all 3 sides. Show that all sides are equal

Distance formula

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